Question: The function $f(x)$ satisfies
\[f(2^x) + xf(2^{-x}) = 1\]for all real numbers $x.$  Find $f(2).$
Solution: Setting $x = 1,$ we get
\[f(2) + f \left( \frac{1}{2} \right) = 1.\]Setting $x = -1,$ we get
\[f \left( \frac{1}{2} \right) - f(2) = 1.\]Subtracting these equations, we get $2f(2) = 0,$ so $f(2) = \boxed{0}.$